Convergence of Approximate and Packet Routing Equilibria to Nash Flows Over Time

TL;DR

The paper addresses whether dynamic traffic equilibria are robust to perturbations and transferable between nonatomic flow-over-time and atomic packet-models. It introduces strict $\delta$-equilibria and proves a unified convergence theorem showing that these perturbed trajectories converge to the exact dynamic equilibrium as $\delta\to 0$, with corollaries for $\varepsilon$-equilibria and packet-model equilibria. The approach relies on embedding packet equilibria into a continuous framework, exploiting thin-flow structures and a careful analysis over generalized subnetworks to control deviations both before and after reaching steady state. The results provide a principled justification for using nonatomic models as predictive tools and enable stability transfer to discrete, packet-based settings, with potential extensions to perturbations of transit times, capacities, and demand. While the bounds are not explicitly tight, the methodology offers a robust route for translating stability insights across dynamic traffic models and perturbations.

Abstract

We consider a dynamic model of traffic that has received a lot of attention in the past few years. Infinitesimally small agents aim to travel from a source to a destination as quickly as possible. Flow patterns vary over time, and congestion effects are modeled via queues, which form based on the deterministic queueing model whenever the inflow into a link exceeds its capacity. Are equilibria in this model meaningful as a prediction of traffic behavior? For this to be the case, a certain notion of stability under ongoing perturbations is needed. Real traffic consists of discrete, atomic ''packets'', rather than being a continuous flow of non-atomic agents. Users may not choose an absolutely quickest route available, if there are multiple routes with very similar travel times. We would hope that in both these situations -- a discrete packet model, with packet size going to 0, and $ε$-equilibria, with $ε$ going to 0 -- equilibria converge to dynamic equilibria in the flow over time model. No such convergence results were known. We show that such a convergence result does hold in single-commodity instances for both of these settings, in a unified way. More precisely, we introduce a notion of ''strict'' $ε$-equilibria, and show that these must converge to the exact dynamic equilibrium in the limit as $ε\to 0$. We then show that results for the two settings mentioned can be deduced from this with only moderate further technical effort.

Figures (1)

• Figure 1: As an example consider a network with only two nodes $s$ and $t$ but four parallel arcs $e_1$ to $e_4$ with transit time $\tau_{e_i}=2i + 1$ and capacities $1$. The network inflow rate is $u_0 = 3$. On the left all hyperplanes are present and the equilibrium trajectory reaches steady state as soon as arcs $e_1$, $e_2$ and $e_3$ are active. To prove \ref{['thm:technical']} we consider inductively also generalized networks with less arcs. In the middle$e_3$ and $e_4$ are removed and therefore $\tilde{E} = \set{e_1, e_2}$ and $E^{\infty} = \emptyset$. We consider an interval $[\theta_0, \theta_1]$ such that all other hyperplanes keep distance to $\ell$. We split the interval at $\theta_{\mathrm{ss}}$ which is the first point in time $\ell$ comes $r_2$ close to the steady-state set $I$. Here, we also illustrated the equilibrium trajectory $\ell^{\bullet}$, which starts within $B_{r_3}(\ell(\theta_{\mathrm{ss}})) \cap I$ and therefore stays in steady state. $\ell^*$ and $\ell^{\bullet}$ are close due to the continuity of equilibrium trajectories; see \ref{['thm:continuity_of_Nash']}. On the right we choose the hyperplanes of $e_2$ and $e_3$, which means that $e_4$ is removed and $e_1$ is promoted to a free arc. Hence $\tilde{E} = \set{e_1, e_2, e_3}$ and $E^{\infty} = \set{e_1}$.

Theorems & Definitions (66)

• Theorem 2.1
• Theorem 2.2
• Theorem 2.3: OSV21
• Theorem 3.1
• Theorem 3.2
• Theorem 3.3
• Definition 3.4
• Theorem 3.5
• Theorem 4.1
• proof : Proof of \ref{['thm:main-full']}